Merge pull request #1 from AndreasFMueller/master

update

23 files changed, 1339 insertions, 0 deletions

diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc
new file mode 100644
index 0000000..c857fec
--- /dev/null
+++ b/vorlesungen/slides/2/Makefile.inc
@@ -0,0 +1,21 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter2 =								\
+	../slides/2/norm.tex						\
+	../slides/2/skalarprodukt.tex					\
+	../slides/2/cauchyschwarz.tex					\
+	../slides/2/polarformel.tex					\
+	../slides/2/funktionenraum.tex					\
+	../slides/2/operatornorm.tex					\
+	../slides/2/linearformnormen.tex				\
+	../slides/2/funktionenalgebra.tex				\
+	../slides/2/frobeniusnorm.tex					\
+	../slides/2/frobeniusanwendung.tex				\
+	../slides/2/quotient.tex					\
+	../slides/2/quotientv.tex					\
+	../slides/2/chapter.tex
+
diff --git a/vorlesungen/slides/2/cauchyschwarz.tex b/vorlesungen/slides/2/cauchyschwarz.tex
new file mode 100644
index 0000000..a24ada8
--- /dev/null
+++ b/vorlesungen/slides/2/cauchyschwarz.tex
@@ -0,0 +1,94 @@
+%
+% cauchyschwarz.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Cauchy-Schwarz-Ungleichung}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz (Cauchy-Schwarz)}
+$\langle\;,\;\rangle$ eine positiv definite, hermitesche Sesquilinearform
+\[
+{\color{darkgreen}
+|\operatorname{Re}\langle u,v\rangle|
+\le
+|\langle u,v\rangle|
+\le
+\|u\|_2\cdot \|v\|_2
+}
+\]
+Gleichheit genau dann, wenn $u$ und $v$ linear abhängig sind
+\end{block}
+\begin{block}{Dreiecksungleichung}
+\vspace{-12pt}
+\begin{align*}
+\|u+v\|_2^2
+&=
+\|u\|_2^2 + 2\operatorname{Re}\langle u,v\rangle + \|v\|_2^2
+\\
+&\le
+\|u\|_2^2 + 2{\color{darkgreen}|\langle u,v\rangle|} + \|v\|_2^2
+\\
+&\le
+\|u\|_2^2 + 2{\color{darkgreen}\|u\|_2\cdot \|v\|_2} + \|v\|_2^2
+\\
+&=(\|u\|_2 + \|v\|_2)^2
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{proof}[Beweis]
+Die quadratische Funktion
+\begin{align*}
+Q(t)
+&=
+\langle u+tv,u+tv\rangle \ge 0
+\\
+\uncover<3->{
+Q(t)
+&=
+\|u\|_2^2 + 2t\operatorname{Re}\langle u,v\rangle + t^2\|v\|_2^2}
+\end{align*}
+\uncover<4->{hat ihr Minimum bei}%
+\begin{align*}
+\uncover<5->{
+t&=
+-\operatorname{Re}\langle u,v\rangle/\|v\|_2^2}
+\intertext{\uncover<6->{mit Wert}}
+\uncover<7->{
+Q(t)
+&=
+\|u\|_2^2
+-2\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<7->{
+&\qquad + \operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<8->{
+0
+&\le
+\|u\|_2^2-\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<9->{
+\operatorname{Re}\langle u,v\rangle^2
+&\le
+\|u\|_2^2\cdot\|v\|_2^2}
+\\
+\uncover<10->{
+\operatorname{Re}\langle u,v\rangle
+&\le
+\|u\|_2\cdot\|v\|_2}
+\qedhere
+\end{align*}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex
new file mode 100644
index 0000000..49e656a
--- /dev/null
+++ b/vorlesungen/slides/2/chapter.tex
@@ -0,0 +1,17 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{2/norm.tex}
+\folie{2/skalarprodukt.tex}
+\folie{2/cauchyschwarz.tex}
+\folie{2/polarformel.tex}
+\folie{2/funktionenraum.tex}
+\folie{2/operatornorm.tex}
+\folie{2/linearformnormen.tex}
+\folie{2/funktionenalgebra.tex}
+\folie{2/frobeniusnorm.tex}
+\folie{2/frobeniusanwendung.tex}
+\folie{2/quotient.tex}
+\folie{2/quotientv.tex}
diff --git a/vorlesungen/slides/2/frobeniusanwendung.tex b/vorlesungen/slides/2/frobeniusanwendung.tex
new file mode 100644
index 0000000..277d600
--- /dev/null
+++ b/vorlesungen/slides/2/frobeniusanwendung.tex
@@ -0,0 +1,80 @@
+%
+% frobeniusanwendung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Anwendung der Frobenius-Norm}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung nach $X\in M_{m\times n}(\mathbb{R})$}
+Die Ableitung $Df=\partial f/\partial X$ der Funktion
+$f\colon M_{m\times n}(\mathbb{R})\to \mathbb{R}$ ist die Matrix
+mit Einträgen
+\begin{align*}
+\biggl(
+\frac{\partial f}{\partial X}
+\biggr)_{ij}
+&=
+\frac{\partial f}{\partial x_{ij}}
+=
+D_{ij}f
+\end{align*}
+\end{block}
+\uncover<2->{%
+\begin{block}{Richtungsableitung}
+\uncover<5->{Die Matrix $Df$ ist ein Gradient:}
+\begin{align*}
+\frac{\partial}{\partial t}f(X+tY)\bigg|_{t=0}
+&=\uncover<3->{
+\sum_{i,j}
+D_{ij} f(X) \cdot y_{ij}}
+\\
+&\uncover<4->{=
+\langle D_{ij}f(X), Y\rangle_F}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Quadratische Minimalprobleme}
+$A=A^t,B,X\in M_n(\mathbb{R})$, Minimum von
+\begin{align*}
+f(X)&=\langle X,AX\rangle_F + \langle B,X\rangle_F
+\intertext{\uncover<7->{Folgerungen:}}
+\uncover<8->{
+\langle X,AY\rangle_F&=\langle AX,Y\rangle_F
+}
+\\
+\uncover<9->{
+D\langle B,\mathstrut\cdot\mathstrut\rangle_F
+&=
+B
+}
+\\
+\uncover<10->{
+D_X\langle X, AY\rangle_F
+&=AY
+}
+\\
+\uncover<11->{
+D_Y\langle X, AY\rangle_F
+&=AX
+}
+\\
+\uncover<12->{
+Df &= 2AX + B
+}
+\intertext{\uncover<13->{Minimum:}}
+\uncover<14->{
+X&=-\frac12 A^{-1}B
+}
+\end{align*}
+\uncover<15->{(Kalman-Filter)}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/frobeniusnorm.tex b/vorlesungen/slides/2/frobeniusnorm.tex
new file mode 100644
index 0000000..461005a
--- /dev/null
+++ b/vorlesungen/slides/2/frobeniusnorm.tex
@@ -0,0 +1,96 @@
+%
+% frobeniusnorm.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Frobenius-Norm}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Skalarprodukt}
+$A,B\in M_{m\times n}(\mathbb{C})$
+\begin{align*}
+\langle A,B\rangle_F
+&\uncover<2->{=
+\sum_{i,j} \overline{a}_{ik}b_{ik}}
+\uncover<3->{=
+\operatorname{Spur} A^*B}
+\\
+\uncover<4->{
+\|A\|_F^2
+&=
+\langle A,A\rangle}
+\uncover<5->{=
+\sum_{i,k} |a_{ik}|^2}
+\end{align*}
+\uncover<6->{%
+$\Rightarrow M_{m\times n}(\mathbb{C})$ ist ein normierter Raum}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Singulärwertzerlegung}
+\vspace{-12pt}
+\begin{align*}
+\uncover<13->{
+A
+&=
+U\Sigma V^*}
+\\
+\uncover<14->{
+A^*A
+&=
+V\Sigma^*U^*U\Sigma V^*}
+\uncover<15->{=
+V\Sigma^*\Sigma V^*}
+\\
+\uncover<16->{%
+\operatorname{Spur}{A^*A}
+&=
+\operatorname{Spur}V\Sigma^*\Sigma V^*}
+\\
+\uncover<17->{%
+&=
+\operatorname{Spur}V^*V\Sigma^*\Sigma}
+\\
+\uncover<18->{%
+&=
+\operatorname{Spur}\Sigma^*\Sigma}
+\uncover<19->{=
+\sum_{i} |\sigma_i|^2}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<7->{%
+\begin{block}{Produkt}
+\vspace{-10pt}
+\begin{align*}
+\|AB\|_F
+\uncover<8->{=
+\sum_{i,j}
+\biggl|
+\sum_{k}
+a_{ik}b_{kj}
+\biggr|^2}
+&\uncover<9->{\le
+\sum_{i,j}
+\biggl(
+\sum_k  |a_{ik}|^2
+\biggr)
+\biggl(
+\sum_l  |b_{lj}|^2
+\biggr)}
+\\
+\uncover<10->{
+&=
+\sum_{i,k} |a_{ik}|^2
+\sum_{l,j} |b_{lj}|^2}
+\uncover<11->{=
+\|A\|_F\cdot \|B\|_F}
+\end{align*}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex
new file mode 100644
index 0000000..9116be4
--- /dev/null
+++ b/vorlesungen/slides/2/funktionenalgebra.tex
@@ -0,0 +1,88 @@
+%
+% funktionenalgebra.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Funktionenalgebra}
+\vspace{-17pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Algebra $C([0,1])$}
+Funktionenraum
+\[
+C([0,1])
+=
+\{f\colon[0,1]\to\mathbb{C}\;|\;\text{$f$ stetig}\}
+\]
+mit Supremum-Norm\uncover<2->{ und punktweisem Produkt
+\[
+(f\cdot g)(x)
+=
+f(x)\cdot g(x)
+\]}
+\end{block}
+\vspace{-8pt}
+\uncover<3->{%
+\begin{block}{Algebranorm}
+\vspace{-12pt}
+\begin{align*}
+\|f\cdot g\|_\infty
+&=
+\sup_{x\in[0,1]} |f(x)g(x)|
+\\
+\uncover<4->{
+&\le
+\sup_{x\in[0,1]}|f(x)|
+\sup_{y\in[0,1]}|g(y)|
+}
+\\
+\uncover<5->{
+&=
+\|f\|_\infty \cdot \|g\|_\infty
+}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Faltungs-Algebra $L^2([0,1])$}
+Funktionenraum
+\[
+L^2=\{f\colon \mathbb{R}\to\mathbb{C}\;|\;\text{$f$ $1$-periodisch}\}
+\]
+mit $L^2$-Skalarprodukt\uncover<7->{ und Faltungsprodukt
+\[
+f*g(x)
+=
+\int_0^1
+\underbrace{f(x-t)}_{(=\gamma_x\check{f})(t)} g(t)\,dx
+\]}
+\end{block}}
+\vspace{-21pt}
+\uncover<8->{%
+\begin{block}{Norm}
+\vspace{-12pt}
+\begin{align*}
+\|f*g\|_2^2
+&\uncover<9->{=\int_0^1 |
+\langle \gamma_x\check{f},g\rangle
+|^2\,dx}
+\\
+\uncover<10->{
+&\le
+\int_0^1
+\|\gamma_t\check{f}\|_2^2
+\|g\|_2^2
+\,dx}
+\\
+\uncover<11->{
+&=\|f\|_2^2\cdot \|g\|_2^2
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/funktionenraum.tex b/vorlesungen/slides/2/funktionenraum.tex
new file mode 100644
index 0000000..f7733cc
--- /dev/null
+++ b/vorlesungen/slides/2/funktionenraum.tex
@@ -0,0 +1,70 @@
+%
+% funktionenraum.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Funktionenraum}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Supremum-Norm}
+Vektorraum
+\[
+C([a,b])
+=
+\{f\colon[a,b]\to\mathbb{R}\;|\; \text{$f$ stetig}\}
+\]
+\only<2->{wird Banachraum }%
+mit der Norm
+\(\displaystyle
+\|f\|
+=
+\|f\|_{\infty}
+=
+\sup_{x\in[a,b]} |f(x)|
+\)
+\end{block}
+\uncover<3->{%
+\begin{block}{$L^1$-Norm}
+Vektorraum
+\[
+L^1([a,b])
+=
+\{f\colon[a,b]\;|\;\text{$f$ integrierbar}\}
+\]
+\only<4->{wird Banachraum }%
+mit der Norm
+\[
+\|f\|_1
+=
+\int_a^b |f(x)|\,dx
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{$L^2$-Norm}
+Vektorraum
+\[
+L^2([a,b])
+=
+\{f\colon[a,b]\to\mathbb{R}\;|\; \|f\|_2^2<\infty\}
+\]
+mit Skalarprodukt
+\begin{align*}
+\langle f,g\rangle
+&=
+\int_a^b \overline{f}(x)g(x)\,dx
+\\
+\|f\|_2^2
+&=
+\int_a^b |f(x)|^2\,dx
+\end{align*}
+\uncover<6->{ist ein Banachraum}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/images/Makefile b/vorlesungen/slides/2/images/Makefile
new file mode 100644
index 0000000..8bce5c9
--- /dev/null
+++ b/vorlesungen/slides/2/images/Makefile
@@ -0,0 +1,32 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	quotient1.jpg quotient2.jpg quotient1.pdf quotient2.pdf
+
+quotient1.png:	quotient1.pov quotient.inc
+	povray +A0.1 +W1920 +H1080 -Oquotient1.png quotient1.pov
+
+quotient1.jpg:	quotient1.png Makefile
+	convert -extract 1360x1040+330+20 quotient1.png \
+		-density 300 -units PixelsPerInch quotient1.jpg
+
+quotient2.png:	quotient2.pov quotient.inc
+	povray +A0.1 +W1920 +H1080 -Oquotient2.png quotient2.pov
+
+quotient2.jpg:	quotient2.png Makefile
+	convert -extract 1360x1040+330+20 quotient2.png \
+		-density 300 -units PixelsPerInch quotient2.jpg
+
+quotient:	quotient.ini quotient.inc quotient.pov
+	rm -rf quotient
+	mkdir quotient
+	povray +A0.1 -Oquotient/0.png -W1920 -H1080 quotient.ini
+
+quotient1.pdf:	quotient1.tex quotient1.jpg
+	pdflatex quotient1.tex
+
+quotient2.pdf:	quotient2.tex quotient2.jpg
+	pdflatex quotient2.tex
+
diff --git a/vorlesungen/slides/2/images/quotient.inc b/vorlesungen/slides/2/images/quotient.inc
new file mode 100644
index 0000000..3fa49d1
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient.inc
@@ -0,0 +1,186 @@
+//
+// quotient.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.035;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+	location <8, 15, -50>
+	look_at <0.4, 0.2, 0.4>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-4, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro kasten()
+	box { <-0.5,-0.5,-0.5>, <1.5,1,1.5> }
+#end
+
+
+arrow(<-0.6,0,0>, <1.6,0,0>, at, White)
+arrow(<0,0,-0.6>, <0,0,1.6>, at, White)
+arrow(<0,-0.6,0>, <0,1.2,0>, at, White)
+
+#declare U = <-1,3,-0.5>;
+#declare V1 = <1,0.2,0>;
+#declare V2 = <0,0.2,1>;
+
+#macro gerade(richtung, farbe)
+	intersection {
+		kasten()
+		cylinder { -U + richtung, U + richtung, at }
+		pigment {
+			color farbe
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#declare A = <0.8, -0.2, 0>;
+#declare B = <0.2, 0.8, 0>;
+
+#macro ebene(vektor1, vektor2)
+#declare n = vcross(vektor1,vektor2);
+
+
+intersection {
+	kasten()
+	plane { n, 0.005 }
+	plane { -n, 0.005 }
+	pigment {
+		color rgbf<0.8,0.8,1,0.7>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+intersection {
+	kasten()
+	union {
+		#declare Xstep = 0.45;
+		#declare X = -5 * Xstep;
+		#while (X < 5.5 * Xstep)
+			cylinder { X*vektor1 - 5*vektor2, X*vektor1 + 5*vektor2, at/2 }
+			#declare X = X + Xstep;
+		#end
+		#declare Ystep = 0.45;
+		#declare Y = -5 * Ystep;
+		#while (Y < 5.5 * Ystep)
+			cylinder { -5*vektor1 + Y*vektor2, 5*vektor1 + Y*vektor2, at/2 }
+			#declare Y = Y + Ystep;
+		#end
+	}
+	pigment {
+		color rgb<0.9,0.9,1>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+
+gerade(O, Red)
+
+#declare gruen = rgb<0.2,0.4,0.2>;
+#declare blau = rgb<0,0.4,0.8>;
+#declare rot = rgb<1,0.4,0.0>;
+
+#macro repraesentanten(vektor1, vektor2) 
+
+#declare d1 = A.x*vektor1 + A.y*vektor2;
+#declare d2 = B.x*vektor1 + B.y*vektor2;
+
+arrow(0, d1 + d2, at, rot)
+gerade(d1 + d2, rot)
+
+gerade(d1, blau)
+arrow(O, d1, at, blau)
+cylinder { d1, d1 + d2, 0.6 * at
+	pigment {
+		color gruen
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+gerade(d2, gruen)
+arrow(O, d2, at, gruen)
+cylinder { d2, d1 + d2, 0.6 * at
+	pigment {
+		color blau
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+#end
+
+#macro vektorraum(s)
+#declare b1 = V1 + s * 0.03 * U;
+#declare b2 = V2 + s * 0.03 * U;
+
+ebene(b1, b2)
+repraesentanten(b1, b2)
+#end
+
diff --git a/vorlesungen/slides/2/images/quotient.ini b/vorlesungen/slides/2/images/quotient.ini
new file mode 100644
index 0000000..f62b21a
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient.ini
@@ -0,0 +1,7 @@
+Input_File_Name="quotient.pov"
+Initial_Frame=0
+Final_Frame=100
+Initial_Clock=-1
+Final_Clock=1
+Cyclic_Animation=off
+Pause_when_Done=off
diff --git a/vorlesungen/slides/2/images/quotient1.jpg b/vorlesungen/slides/2/images/quotient1.jpg
new file mode 100644
index 0000000..aeb713e
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.jpg
diff --git a/vorlesungen/slides/2/images/quotient1.pov b/vorlesungen/slides/2/images/quotient1.pov
new file mode 100644
index 0000000..60bab7f
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.pov
@@ -0,0 +1,8 @@
+//
+// quotient1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "quotient.inc"
+
+vektorraum(-1)
diff --git a/vorlesungen/slides/2/images/quotient1.tex b/vorlesungen/slides/2/images/quotient1.tex
new file mode 100644
index 0000000..30d82d2
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.tex
@@ -0,0 +1,29 @@
+%
+% quotient1.tex -- Vektorraumquotient
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.7,0,0}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\node at (0,0) {\includegraphics[width=8cm]{quotient1.jpg}};
+
+\node[color=blue] at (0.7,-1.3) {$v$};
+\node[color=darkgreen] at (-1.0,0.1) {$w$};
+\node[color=orange] at (2.5,0.1) {$v+w$};
+\node[color=darkred] at (-2.1,-0.9) {$0$};
+\node[color=darkred] at (-3.1,2.4) {$U$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/vorlesungen/slides/2/images/quotient2.jpg b/vorlesungen/slides/2/images/quotient2.jpg
new file mode 100644
index 0000000..345cf22
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.jpg
diff --git a/vorlesungen/slides/2/images/quotient2.pov b/vorlesungen/slides/2/images/quotient2.pov
new file mode 100644
index 0000000..771425d
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.pov
@@ -0,0 +1,8 @@
+//
+// quotient2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "quotient.inc"
+
+vektorraum(1)
diff --git a/vorlesungen/slides/2/images/quotient2.tex b/vorlesungen/slides/2/images/quotient2.tex
new file mode 100644
index 0000000..607fd03
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.tex
@@ -0,0 +1,29 @@
+%
+% quotient2.tex -- Vektorraumquotient
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.7,0,0}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\node at (0,0) {\includegraphics[width=8cm]{quotient2.jpg}};
+
+\node[color=blue] at (0.57,-0.94) {$v$};
+\node[color=darkgreen] at (-1.15,0.65) {$w$};
+\node[color=orange] at (2.15,1) {$v+w$};
+\node[color=darkred] at (-2.1,-0.9) {$0$};
+\node[color=darkred] at (-3.1,2.4) {$U$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/vorlesungen/slides/2/linearformnormen.tex b/vorlesungen/slides/2/linearformnormen.tex
new file mode 100644
index 0000000..8993f66
--- /dev/null
+++ b/vorlesungen/slides/2/linearformnormen.tex
@@ -0,0 +1,76 @@
+%
+% linearformnormen.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Linearformen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Linearformen $\varphi\colon L^1\to\mathbb{R}$}
+Beispiel: $g\in C([a,b])$
+\[
+\varphi(f)
+=
+\int_a^b g(x)f(x)\,dx
+\]
+\uncover<2->{%
+erfüllt
+\begin{align*}
+|\varphi(f)|
+&=
+\biggl|\int_a^b g(x)f(x)\,dx\biggr|
+\\
+\uncover<3->{
+&\le \|g\|_\infty\cdot \|f\|_1
+}
+\end{align*}}
+\uncover<4->{%
+und hat daher die Operatornorm
+\[
+\|\varphi\|_{C([a,b])^*}
+=
+\|g\|_\infty
+\]}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Linearformen $\varphi\colon L^2\to\mathbb{R}$}
+\uncover<5->{%
+Darstellungssatz von Riesz: $\exists g\in L^2$
+\[
+\varphi(f) = \langle g,f\rangle
+\]}
+\uncover<6->{%
+erfüllt Cauchy-Schwarz}
+\begin{align*}
+\uncover<7->{
+|\varphi(f)|
+&=
+|\langle g,f\rangle|}
+\\
+\uncover<8->{
+&\le
+\|g\|_2 \cdot \|f\|_2
+}
+\end{align*}
+\uncover<9->{%
+und hat daher die Operatornorm
+\[
+\|\varphi\|_{L^2([a,b])^*}
+= \|g\|_2
+\]}
+\end{block}
+\end{column}
+\end{columns}
+
+\vspace{8pt}
+{\usebeamercolor[fg]{title}
+\uncover<10->{%
+$\Rightarrow$
+Operatornorm hängt von den Vektorraumnormen ab}
+}
+\end{frame}
diff --git a/vorlesungen/slides/2/norm.tex b/vorlesungen/slides/2/norm.tex
new file mode 100644
index 0000000..35d2513
--- /dev/null
+++ b/vorlesungen/slides/2/norm.tex
@@ -0,0 +1,58 @@
+%
+% norm.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Norm}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Wozu}
+Ziel: Konvergenz von Folgen, Grenzwert in einem Vektorraum
+\end{block}
+\uncover<7->{%
+\begin{block}{Cauchy-Folge}
+Eine Folge $(x_n)_{n\in\mathbb{N}}$ von Vektoren in $V$ heisst
+{\em Cauchy-Folge},
+wenn es für alle $\varepsilon >0$ ein $N$ gibt mit
+\[
+\|x_n-x_m\| < \varepsilon\; \forall n,m>N
+\]
+\end{block}}
+\vspace{-8pt}
+\uncover<8->{%
+\begin{block}{Grenzwert}
+$x\in V$ heisst Grenzwert der Folge $x_n$, wenn es für alle $\varepsilon>0$
+ein $N$ gibt mit
+\[
+\| x-x_n\| < \varepsilon \;\forall n>N
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Definition}
+$V$ ein $\mathbb{R}$-Vektorraum.
+Eine Funktion
+\[
+\|\cdot\| \colon V \to \mathbb{R}_{\ge 0} : v \mapsto \|v\|
+\]
+heisst eine {\em Norm}, wenn
+\begin{itemize}
+\item<3-> $\| v \|>0$ für $v\ne 0$
+\item<4-> $\|\lambda v\| = |\lambda|\cdot\|v\|$
+\item<5-> $\| u + v \| \le \|u\| + \|v\|$ (Dreiecksungleichung)
+\end{itemize}
+\uncover<6->{%
+Ein Vektorraum mit einer Norm heisst {\em normierter Raum}}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Banach-Raum}
+Normierter Raum, in dem jede Cauchy-Folge einen Grenwzert hat
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/operatornorm.tex b/vorlesungen/slides/2/operatornorm.tex
new file mode 100644
index 0000000..d20461a
--- /dev/null
+++ b/vorlesungen/slides/2/operatornorm.tex
@@ -0,0 +1,59 @@
+%
+% operatorname.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Operatornorm}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Lineare Operatoren}
+$A\colon U\to V$ lineare Abbildung mit $U$, $V$ normiert
+\end{block}}
+\uncover<3->{%
+\begin{block}{Operatornorm}
+eines linearen Operators $A$:
+\[
+\|A\| 
+=
+\sup_{\|x\|_U\le 1} \|Ax\|_V
+\]
+\uncover<4->{$\Rightarrow \|Ax\| \le \| A \|\cdot \|x\|$}
+\end{block}}
+\uncover<5->{%
+\begin{block}{Stetigkeit}
+Wenn $\|A\|<\infty$, dann ist $A$ stetig, d.~h.
+\[
+\lim_{n\to\infty} Ax_n
+=
+A\lim_{n\to\infty} x_n
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Algebranorm}
+$A$ ein normierter Raum, der auch ein Algebra ist.
+Dann heisst $A$ eine normierte Algebra, wenn
+\[
+\| ab\| \le \| a\|\cdot \|b\|
+\quad\forall a,b\in A
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Operatoralgebra}
+$U$ ein normierter Raum, dann ist die Algebra der linearen Operatoren
+$A\colon U\to U$ mit der Operatornorm eine normierte Algebra
+\end{block}}
+\uncover<8->{%
+\begin{block}{Banach-Algebra}
+Ein Banach-Raum, der auch eine normierte Algebra ist
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/polarformel.tex b/vorlesungen/slides/2/polarformel.tex
new file mode 100644
index 0000000..ebdbf81
--- /dev/null
+++ b/vorlesungen/slides/2/polarformel.tex
@@ -0,0 +1,113 @@
+%
+% polarformel.tex
+%
+% (c) 2021 Prod Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkcolor}{rgb}{0,0.6,0}
+\def\yone{-2.1}
+\def\ytwo{-3.55}
+\def\ythree{-5.0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Polarformel}
+\vspace{-5pt}
+\begin{block}{Aufgabe}
+$\langle x,y\rangle$ aus Werten von $\|\cdot\|_2$ rekonstruieren:
+
+\end{block}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\node at (0,0) {$
+\begin{aligned}
+\uncover<2->{
+\|x+ty\|_2^2
+&=
+\|x\|_2^2
++t\langle x,y\rangle
++\overline{t}\langle y,x\rangle
++ \|y\|_2^2}
+\\
+\uncover<3->{
+&=
+\|x\|_2^2
++t\langle x,y\rangle
++\overline{t\langle x,y\rangle}
++ \|y\|_2^2}
+\\
+\uncover<4->{
+&=
+\|x\|_2^2
++2\operatorname{Re}(t\langle x,y\rangle)
++ \|y\|_2^2}
+\end{aligned}$};
+
+\uncover<5->{
+	\draw[->] (-1,-0.9) -- (-3.3,{\yone+0.25});
+	\node at (-3.5,\yone) {$
+	\|x\pm y\|_2^2
+	=
+	\|x\|_2^2
+	\pm2\operatorname{Re}\langle x,y\rangle
+	+
+	\|y\|_2^2
+	$};
+}
+
+\uncover<8->{
+	\draw[->] (1,-0.9) -- (3.3,{\yone+0.25});
+	\node at (3.5,\yone) {$
+	\|x\pm iy\|_2^2
+	=
+	\|x\|_2^2
+	\pm2i\operatorname{Im}\langle x,y\rangle
+	+
+	\|y\|_2^2
+	$};
+}
+
+\uncover<6->{
+	\draw[->] (-3.5,{\yone-0.2}) -- (-3.5,{\ytwo+0.2});
+	\node at (-3.5,\ytwo) {$\operatorname{Re}\langle x,y\rangle
+	=
+	\frac12\bigl(
+	\|x+y\|_2^2-\|x-y\|_2^2
+	\bigr)
+	$};
+}
+
+\uncover<9->{
+	\draw[->] (3.5,{\yone-0.2}) -- (3.5,{\ytwo+0.2});
+	\node at (3.5,\ytwo) {$
+	\operatorname{Im}\langle x,y\rangle
+	=
+	\frac1{2i}\bigl(
+	\|x+iy\|_2^2-\|x-iy\|_2^2
+	\bigr)
+	$};
+}
+
+\uncover<7->{
+	\draw[->] (-3.3,{\ytwo-0.25}) -- (-1.5,{\ythree+0.25});
+	\node at (0,\ythree) {$
+	\langle x,y\rangle
+	=
+	\frac12\bigl(
+	\|x+y\|_2^2-\|x-y\|_2^2
+	\uncover<10->{
+		+
+		\|x+iy\|_2^2-\|x-iy\|_2^2
+	}
+	\bigr)$};
+}
+
+\uncover<10->{
+	\draw[->] (3.3,{\ytwo-0.25}) -- (1.5,{\ythree+0.25});
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/quotient.tex b/vorlesungen/slides/2/quotient.tex
new file mode 100644
index 0000000..24b0523
--- /dev/null
+++ b/vorlesungen/slides/2/quotient.tex
@@ -0,0 +1,110 @@
+%
+% quotient.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.7,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\s{0.3}
+\def\punkt#1#2{({#1-3*#2},{8*#2})}
+\def\gerade#1{
+\draw[darkgreen,line width=1.4pt]
+	\punkt{#1}{1}
+	--
+	\punkt{#1}{-1};
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quotientenraum}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Einen Unterraum ``ignorieren''}
+{\usebeamercolor[fg]{title}Gegeben:} $U\subset V$ ein Unterraum
+\\
+{\usebeamercolor[fg]{title}Gesucht:} Eine Projektion auf einen Vektorraum,
+in dem die Richtungen in $U$ zu $0$ gemacht werden
+\end{block}
+\uncover<2->{%
+\begin{block}{Projektion}
+In $V$ Klassen bilden:
+\[
+\pi
+\colon
+v\mapsto
+\llbracket v\rrbracket
+=
+v+U
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<3->{%
+\begin{block}{Quotientenraum}
+\vspace{-12pt}
+\begin{align*}
+V/U
+&=
+\{ v+U\;|\; v\in V \}
+\\
+\uncover<4->{\pi(\lambda v)&=\lambda v+U= \lambda \pi(v)}
+\\
+\uncover<5->{\pi(v+w)
+&=
+v+w+U}
+\ifthenelse{\boolean{presentation}}{
+\only<6>{=
+v+U+w+U}}{}
+\uncover<7->{=
+\pi(v) + \pi(w)}
+\phantom{blubb}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (U) at (-3,8);
+\def\t{0.03}
+\begin{scope}
+\clip (-2,-2) rectangle (4,4.8);
+\draw[color=darkred,line width=2pt] (-3,8) -- (1.5,-4);
+\node[color=darkred] at (-1.45,4.6) {$U$};
+\node[color=darkred] at (-0.05,-0.05) [above left] {$0$};
+
+\gerade{2.5}
+
+\ifthenelse{\boolean{presentation}}{
+	\foreach \n in {8,...,25}{
+		\pgfmathparse{(\n-12)*0.04}
+		\xdef\s{\pgfmathresult}
+		\only<\n>{
+			\draw[color=blue,line width=1.2pt]
+				\punkt{-5}{-2*\s} -- \punkt{5}{2*\s};
+			\draw[->,color=blue,line width=2pt]
+				(0,0) -- \punkt{2.5}{\s};
+			\node[color=blue] at \punkt{2.5}{\s}
+				[above right] {$v'$};
+		}
+	}
+}{
+	\xdef\s{0.35}
+	\draw[color=blue,line width=1.2pt]
+		\punkt{-5}{-2*\s} -- \punkt{5}{2*\s};
+	\draw[->,color=blue,line width=2pt] (0,0) -- \punkt{2.5}{\s};
+	\node[color=blue] at \punkt{2.5}{\s} [above right] {$v'$};
+}
+
+\draw[->,color=darkgreen,line width=1.4pt] (0,0) -- \punkt{2.5}{0.1};
+
+\node[color=darkgreen] at \punkt{2.5}{0.1} [above right] {$v$};
+
+\end{scope}
+\draw[->] (-2,0) -- (4,0) coordinate[label={$x$}];
+\draw[->] (0,-2) -- (0,5) coordinate[label={right:$x$}];
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/quotientv.tex b/vorlesungen/slides/2/quotientv.tex
new file mode 100644
index 0000000..dc01f21
--- /dev/null
+++ b/vorlesungen/slides/2/quotientv.tex
@@ -0,0 +1,62 @@
+%
+% quotientv.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.7,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Quotient}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.33\textwidth}
+\begin{block}{Repräsentanten}
+Jeder Unterraum $W\subset V$ mit 
+$W\cap U = \{0\}$
+kann als Menge von Repräsentanten
+für
+\begin{align*}
+V/U
+&=
+\{v+U\;|\;v \in V\}
+\\
+&\simeq W
+\end{align*}
+dienen.
+\end{block}
+\uncover<3->{%
+\begin{block}{Orthogonalraum}
+Mit Skalarprodukt ist
+$W=U^\perp$ eine bevorzugte Wahl
+\end{block}}
+\end{column}
+\begin{column}{0.66\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\only<1>{
+	\node at (0,0)
+	{\includegraphics[width=8.5cm]{../slides/2/images/quotient1.jpg}};
+	\node[color=darkgreen] at (-0.5,0.3) {$v$};
+	\node[color=blue] at (0.7,-1.4) {$w$};
+	\node[color=orange] at (2.7,0.1) {$v+w$};
+	\fill[color=white,opacity=0.5] (3.7,1.0) circle[radius=0.25];
+	\node at (3.7,1.0) {$W$};
+}
+\only<2->{
+	\node at (0,0)
+	{\includegraphics[width=8.5cm]{../slides/2/images/quotient2.jpg}};
+	\node[color=darkgreen] at (-0.75,0.95) {$v$};
+	\node[color=blue] at (0.6,-1.05) {$w$};
+	\node[color=orange] at (2.36,1.05) {$v+w$};
+	\fill[color=white,opacity=0.5] (3.7,2.9) circle[radius=0.25];
+	\node at (3.7,2.9) {$W$};
+}
+\node[color=darkred] at (-3.3,2.6) {$U$};
+\node[color=darkred] at (-2.25,-1.0) {$0$};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex
new file mode 100644
index 0000000..99d8a73
--- /dev/null
+++ b/vorlesungen/slides/2/skalarprodukt.tex
@@ -0,0 +1,96 @@
+%
+% skalarprodukt.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Skalarprodukt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Positiv definite, symmetrische Bilinearform}
+$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$
+\begin{itemize}
+\item<2->
+Bilinear:
+\begin{align*}
+\langle \alpha u+\beta v,w\rangle
+&=
+\alpha\langle u,w\rangle
++
+\beta\langle v,w\rangle
+\\
+\langle u,\alpha v+\beta w\rangle
+&=
+\alpha\langle u,v\rangle
++
+\beta\langle u,w\rangle
+\end{align*}
+\item<3->
+Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$
+\item<4->
+$\langle x,x\rangle >0 \quad\forall x\ne 0$
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Positive definite, hermitesche Sesquilinearform}
+$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$
+\begin{itemize}
+\item<6->
+Sesquilinear:
+\begin{align*}
+\langle \alpha u+\beta v,w\rangle
+&=
+\overline{\alpha}\langle u,w\rangle
++
+\overline{\beta}\langle v,w\rangle
+\\
+\langle u,\alpha v+\beta w\rangle
+&=
+\alpha\langle u,v\rangle
++
+\beta\langle u,w\rangle
+\end{align*}
+\item<7->
+Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$
+\item<8->
+$\langle x,x\rangle >0 \quad\forall x\ne 0$
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.28\textwidth}
+\uncover<9->{%
+\begin{block}{$2$-Norm}
+$\|v\|_2^2 = \langle v,v\rangle$
+\\
+$\|v\|_2 = \sqrt{\langle v,v\rangle}$
+\end{block}}
+\end{column}
+\begin{column}{0.78\textwidth}
+\uncover<10->{%
+\begin{itemize}
+\item<11-> $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$
+\item<12-> $\| \lambda v \|_2
+=
+\sqrt{\langle \lambda v,\lambda v\rangle\mathstrut}
+=
+\sqrt{\overline{\lambda}\lambda\langle v,v\rangle}
+=
+|\lambda|\cdot \|v\|_2$
+\item<13->
+\raisebox{-8pt}{
+$\begin{aligned}
+\|u+v\|_2^2 &= \|u\|_2^2 + 2{\color{red}\operatorname{Re}\langle u,v\rangle} + \|v\|_2^2
+\\
+(\|u\|_2+\|v\|_2)^2 &= \|u\|_2^2 + 2{\color{red}\|u\|_2\|v\|_2} + \|v\|_2^2
+\end{aligned}$}
+\end{itemize}}
+\end{column}
+\end{columns}
+\end{frame}